Etch bias characterization and method of using the same

ABSTRACT

A method involving determining an etch bias for a pattern to be etched using an etch step of a patterning process based on an etch bias model, the etch bias model including a formula having a variable associated with a spatial property of the pattern or with an etch plasma species concentration of the etch step, and including a mathematical term including a natural exponential function to the power of a parameter that is fitted or based on an etch time of the etch step; and adjusting the patterning process based on the determined etch bias.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is based upon and claims priority of U.S. ProvisionalApplication No. 62/463,556, filed Feb. 24, 2017, and entitled “Etch BiasCharacterization and Method of Using the Same,” the disclosure of whichis incorporated herein by reference in its entirety.

FIELD

The invention relates generally to device manufacturing and methods ofconfiguring and evaluating device manufacturing processes.

BACKGROUND

A lithography apparatus is a machine that applies a desired pattern ontoa target portion of a substrate. Lithography apparatus can be used, forexample, in the manufacture of integrated circuits (ICs). In thatcircumstance, a patterning device, which is alternatively referred to asa mask or a reticle, may be used to generate a pattern corresponding toan individual layer of the IC, and this pattern can be imaged onto atarget portion (e.g. comprising part of, one or several dies) on asubstrate (e.g. a silicon wafer) that has a layer of radiation-sensitivematerial (resist such as photoresist). In general, a single substratewill contain a network of adjacent target portions that are successivelyexposed. Known lithography apparatus include so-called steppers, inwhich each target portion is irradiated by exposing an entire patternonto the target portion in one go, and so-called scanners, in which eachtarget portion is irradiated by scanning the pattern through the beam ina given direction (the “scanning”-direction) while synchronouslyscanning the substrate parallel or anti parallel to this direction.

Prior to transferring the pattern from the patterning device to thesubstrate, the substrate may undergo various procedures, such aspriming, resist coating and a soft bake. After exposure, the substratemay be subjected to other procedures, such as a post-exposure bake(PEB), development, a hard bake and measurement/inspection of thetransferred pattern. This array of procedures is used as a basis to makean individual layer of a device, e.g., an IC. The substrate may thenundergo various processes such as etching, ion-implantation (doping),metallization, oxidation, chemo-mechanical polishing, etc., all intendedto finish off the individual layer of the device. If several layers arerequired in the device, then the whole procedure, or a variant thereof,is repeated for each layer. Eventually, a device will be present in eachtarget portion on the substrate. These devices are then separated fromone another by a technique such as dicing or sawing, whence theindividual devices can be mounted on a carrier, connected to pins, etc.While the term substrate encompasses an underlying base (e.g., silicon),it can also, where applicable, encompass one or more layers overlyingthe base. Thus, transferring a pattern into or onto the substrate caninclude transfer of the pattern onto one or more layers on thesubstrate.

Thus, manufacturing devices, such as semiconductor devices, typicallyinvolves processing a substrate (e.g., a semiconductor wafer) using anumber of fabrication processes to form various features and multiplelayers of the devices. Such layers and features are typicallymanufactured and processed using, e.g., deposition, lithography, etch,chemical-mechanical polishing, and ion implantation. Multiple devicesmay be fabricated on a plurality of dies on a substrate and thenseparated into individual devices. This device manufacturing process maybe considered a patterning process. A patterning process involves apatterning step, such as optical and/or nanoimprint lithography using apatterning device in a lithographic apparatus, to transfer a pattern onthe patterning device to a substrate and typically, but optionally,involves one or more related pattern processing steps, such as resistdevelopment by a development apparatus, baking of the substrate using abake tool, etching using the pattern using an etch apparatus, etc.

SUMMARY

Etch bias of patterning processes may cause deviation of a devicepattern from a set of target dimensions of the device pattern.Determining etch bias may be computationally complex because of, e.g.,the complexity of the device pattern, the scale of the device pattern,the chemical processes that may occur in an etch process that transfersa device pattern from a layer of patterning material to a substrate,and/or a coating, transient or otherwise, that may form on sidewalls ofa substrate as a device pattern is etched into the device pattern.

Thus, there is provided one or more methods of determining a change in adevice pattern dimension during an etch process that transfers a devicepattern in a layer on a substrate to the substrate.

Aspects of the present disclosure relate to collecting a set of valuesfor one or more spatial properties of a device pattern, such as one ormore dimensions, one or more positions of a part (e.g., an edge) of adevice pattern, etc., at a plurality of sites of a device formed byetching, using a computing device to fit a mathematical model having aset of one or more fitting parameters to the set of spatial properties,calculating, based on the parameterized model, an etch bias for thedevice pattern at at least one location thereof for an etch process,wherein the model comprises a formula that includes a variableassociated with a spatial property of the device pattern at the locationand/or a plasma species concentration, and a mathematical termassociated with a natural exponential function to the power of aparameter that is based on (e.g., a function) of etch time for the etchprocess, and adjusting a patterning process based on the calculated etchbias (e.g., adjusting a border of a region of a patterning device usedto form the device pattern as part of the patterning process).

In an embodiment, there is provided a method, comprising: determining,by a hardware computer, an etch bias for a pattern to be etched using anetch step of a patterning process based on an etch bias model, the etchbias model comprising a formula including a variable associated with aspatial property of the pattern or with an etch plasma speciesconcentration of the etch step, and including a mathematical termcomprising a natural exponential function to the power of a parameterthat is fitted or based on an etch time of the etch step; and adjustingthe patterning process based on the determined etch bias.

In an embodiment, there is provided a method, comprising: determining,by a hardware computer, an etch bias for a pattern to be etched using anetch step of a patterning process based on an etch bias model, the etchbias model comprising a function of an etch plasma species concentrationand a patterning material concentration; and adjusting the patterningprocess based on the determined etch bias.

In an embodiment, there is provided a computer program productcomprising a non-transitory computer readable medium having instructionsrecorded thereon, the instructions when executed by a computerimplementing a method as described herein.

These and other features of the present invention, as well as themethods of operation and functions of the related elements of structureand the combination of parts and economies of manufacture, will becomemore apparent upon consideration of the following description and theappended claims with reference to the accompanying drawings, all ofwhich form a part of this specification, wherein like reference numeralsdesignate corresponding parts in the various figures. It is to beexpressly understood, however, that the drawings are for the purpose ofillustration and description only and are not intended as a definitionof the limits of the invention. As used in the specification and in theclaims, the singular form of “a”, “an”, and “the” include pluralreferents unless the context clearly dictates otherwise. In addition, asused in the specification and the claims, the term “or” means “and/or”unless the context clearly dictates otherwise.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention is illustrated by way of example, and not by way oflimitation, in the figures of the accompanying drawing and in which likereference numerals refer to similar elements.

FIG. 1 depicts a schematic diagram of an embodiment of a lithographicapparatus;

FIG. 2 depicts a schematic diagram of an embodiment of a lithographiccell;

FIG. 3 depicts a schematic diagram of an embodiment of a modeled area ofa device;

FIG. 4 depicts a flow diagram of an embodiment of a method fordetermining etch bias;

FIG. 5 a cross-sectional diagram of an embodiment of a device featureand associated pattern element during an etch process;

FIG. 6 depicts a schematic diagram of a modeled area of an embodiment ofa device;

FIG. 7 depicts a schematic diagram of a modeled area of an embodiment ofa device; and

FIG. 8 depicts a schematic diagram of a modeled area of an embodiment ofa device.

FIG. 9 depicts a block diagram that illustrates an embodiment of acomputer system.

DETAILED DESCRIPTION

FIG. 1 schematically depicts a lithographic apparatus LA in associationwith which the techniques described herein can be utilized. Theapparatus includes an illumination optical system (illuminator) ILconfigured to condition a radiation beam B (e.g., ultraviolet (UV), deepultraviolet (DUV) or extreme ultraviolet (EUV) radiation), a patterningdevice support or support structure (e.g., a mask table) MT constructedto support a patterning device (e.g., a mask) MA and connected to afirst positioner PM configured to accurately position the patterningdevice in accordance with certain parameters; one or more substratetables (e.g., a wafer table) WTa, WTb constructed to hold a substrate(e.g., a resist coated wafer) W and connected to a second positioner PWconfigured to accurately position the substrate in accordance withcertain parameters; and a projection optical system (e.g., a refractive,reflective, catoptric or catadioptric optical system) PS configured toproject a pattern imparted to the radiation beam B by patterning deviceMA onto a target portion C (e.g., including one or more dies) of thesubstrate W.

The illumination optical system may include various types of opticalcomponents, such as refractive, reflective, magnetic, electromagnetic,electrostatic or other types of optical components, or any combinationthereof, for directing, shaping, or controlling radiation. In thisparticular case, the illumination system also comprises a radiationsource SO.

The patterning device support holds the patterning device in a mannerthat depends on the orientation of the patterning device, the design ofthe lithographic apparatus, and other conditions, such as for examplewhether or not the patterning device is held in a vacuum environment.The patterning device support can use mechanical, vacuum, electrostaticor other clamping techniques to hold the patterning device. Thepatterning device support may be a frame or a table, for example, whichmay be fixed or movable as required. The patterning device support mayensure that the patterning device is at a desired position, for examplewith respect to the projection system. Any use of the terms “reticle” or“mask” herein may be considered synonymous with the more general term“patterning device.”

The term “patterning device” used herein should be broadly interpretedas referring to any device that can be used to impart a radiation beamwith a pattern in its cross-section such as to create a pattern in atarget portion of the substrate. It should be noted that the patternimparted to the radiation beam may not exactly correspond to the desiredpattern in the target portion of the substrate, for example if thepattern includes phase-shifting features or so called assist features.Generally, the pattern imparted to the radiation beam will correspond toa particular functional layer in a device being created in the targetportion, such as an integrated circuit.

The patterning device may be transmissive or reflective. Examples ofpatterning devices include masks, programmable minor arrays, andprogrammable LCD panels. Masks are well known in lithography, andinclude mask types such as binary, alternating phase-shift, andattenuated phase-shift, as well as various hybrid mask types. An exampleof a programmable mirror array employs a matrix arrangement of smallmirrors, each of which can be individually tilted so as to reflect anincoming radiation beam in different directions. The tilted mirrorsimpart a pattern in a radiation beam, which is reflected by the mirrormatrix. As another example the patterning device comprises a LCD matrix.

As here depicted, the apparatus is of a transmissive type (e.g.,employing a transmissive patterning device). However, the apparatus maybe of a reflective type (e.g., employing a programmable minor array of atype as referred to above, or employing a reflective mask (e.g., for anEUV system)).

The lithographic apparatus may also be of a type wherein at least aportion of the substrate may be covered by a liquid having a relativelyhigh refractive index, e.g., water, so as to fill a space between theprojection system and the substrate. An immersion liquid may also beapplied to other spaces in the lithographic apparatus, for example,between the mask and the projection system. Immersion techniques arewell known in the art for increasing the numerical aperture ofprojection systems. The term “immersion” as used herein does not meanthat a structure, such as a substrate, must be submerged in liquid, butrather only means that liquid is located between the projection systemand the substrate during exposure.

Referring to FIG. 1, the illuminator IL receives a radiation beam from aradiation source SO (e.g., a mercury lamp or excimer laser, LPP (laserproduced plasma) EUV source). The source and the lithographic apparatusmay be separate entities, for example when the source is an excimerlaser. In such cases, the source is not considered to form part of thelithographic apparatus and the radiation beam is passed from the sourceSO to the illuminator IL with the aid of a beam delivery system BDincluding, for example, suitable directing mirrors and/or a beamexpander. In other cases the source may be an integral part of thelithographic apparatus, for example when the source is a mercury lamp.The source SO and the illuminator IL, together with the beam deliverysystem BD if required, may be referred to as a radiation system.

The illuminator IL may include an adjuster AD configured to adjust thespatial and/or angular intensity distribution of the radiation beam.Generally, at least the outer and/or inner radial extent (commonlyreferred to as σ-outer and σ-inner, respectively) of the intensitydistribution in a pupil plane of the illuminator can be adjusted. Inaddition, the illuminator IL may include various other components, suchas an integrator IN and a condenser CO. The illuminator may be used tocondition the radiation beam, to have a desired uniformity and intensitydistribution in its cross section.

The radiation beam B is incident on the patterning device (e.g., mask)MA, which is held on the patterning device support (e.g., mask table)MT, and is patterned by the patterning device. Having traversed thepatterning device (e.g., mask) MA, the radiation beam B passes throughthe projection optical system PS, which focuses the beam onto a targetportion C of the substrate W, thereby projecting an image of the patternon the target portion C. With the aid of the second positioner PW andposition sensor IF (e.g., an interferometric device, linear encoder, 2-Dencoder or capacitive sensor), the substrate table WT can be movedaccurately, e.g., so as to position different target portions C in thepath of the radiation beam B. Similarly, the first positioner PM andanother position sensor (which is not explicitly depicted in FIG. 1) canbe used to accurately position the patterning device (e.g., mask) MAwith respect to the path of the radiation beam B, e.g., after mechanicalretrieval from a mask library, or during a scan.

Patterning device (e.g., mask) MA and substrate W may be aligned usingpatterning device alignment marks M1, M2 and substrate alignment marksP1, P2. Although the substrate alignment marks as illustrated occupydedicated target portions, they may be located in spaces between targetportions (these are known as scribe-lane alignment marks). Similarly, insituations in which more than one die is provided on the patterningdevice (e.g., mask) MA, the patterning device alignment marks may belocated between the dies. Small alignment markers may also be includedwithin dies, in amongst the device features, in which case it isdesirable that the markers be as small as possible and not require anydifferent imaging or process conditions than adjacent features. Thealignment system, which detects the alignment markers, is describedfurther below.

Lithographic apparatus LA in this example is of a so-called dual stagetype which has two substrate tables WTa, WTb and two stations—anexposure station and a measurement station—between which the substratetables can be exchanged. While one substrate on one substrate table isbeing exposed at the exposure station, another substrate can be loadedonto the other substrate table at the measurement station and variouspreparatory steps carried out. The preparatory steps may include mappingthe surface control of the substrate using a level sensor LS, measuringthe position of alignment markers on the substrate using an alignmentsensor AS, performing any other type of metrology or inspection, etc.This enables a substantial increase in the throughput of the apparatus.More generally, the lithography apparatus may be of a type having two ormore tables (e.g., two or more substrate tables, a substrate table and ameasurement table, two or more patterning device tables, etc.). In such“multiple stage” devices a plurality of the multiple tables may be usedin parallel, or preparatory steps may be carried out on one or moretables while one or more other tables are being used for exposures.

While a level sensor LS and an alignment sensor AS are shown adjacentsubstrate table WTb, it will be appreciated that, additionally oralternatively, a level sensor LS and an alignment sensor AS can beprovided adjacent the projection system PS to measure in relation tosubstrate table WTa.

The depicted apparatus can be used in a variety of modes, including forexample a step mode or a scan mode. The construction and operation oflithographic apparatus is well known to those skilled in the art andneed not be described further for an understanding of embodiments of thepresent invention.

As shown in FIG. 2, the lithographic apparatus LA forms part of alithographic system, referred to as a lithographic cell LC or alithocell or cluster. The lithographic cell LC may also includeapparatus to perform pre- and post-exposure processes on a substrate.Conventionally these include spin coaters SC to deposit resist layers,developers DE to develop exposed resist, chill plates CH and bake platesBK. A substrate handler, or robot, RO picks up substrates frominput/output ports I/O1, I/O2, moves them between the different processapparatus and delivers then to the loading bay LB of the lithographicapparatus. These devices, which are often collectively referred to asthe track, are under the control of a track control unit TCU which isitself controlled by the supervisory control system SCS, which alsocontrols the lithographic apparatus via lithography control unit LACU.Thus, the different apparatus can be operated to maximize throughput andprocessing efficiency.

In an embodiment of a patterning process, a device pattern may betransferred from a patterning device into a layer of patterning material(e.g., resist) on a substrate (e.g., a semiconductor substrate). Adevice pattern in a layer of patterning material may be transferred tomaterial under the patterning material by a pattern transfer process. Inan embodiment, the device pattern may be transferred to a substrate by asubstrate material removal process such as etching. In an embodiment,the etching comprises plasma etching. In an embodiment, plasma etchinginvolves creation of ionized and non-ionized chemical species in alow-pressure environment adjacent to a surface of a substrate. Plasmaetching may involve accelerating some chemical species onto a surface ofthe substrate in order to promote removal of substrate material.Chemical bonds between atoms of a substrate may be weakened by achemical reaction of some of the atoms to chemical species originatingin the plasma. Removal of substrate material atoms during an etchingprocess may be enhanced by transfer of kinetic energy to substratematerial by plasma species that accelerate toward the substrate materialand strike the substrate material, promoting vibrational motion ofsubstrate atoms with respect to neighboring substrate material atoms andwith respect to atoms and/or molecules originating from the plasma.Atoms of substrate material may be chemically bonded to atoms/moleculesthat originate in an etching plasma after vibrational energy transferredto the substrate material breaks one or more bonds between substratematerial atoms, while bonds between a plasma species and a liberatedsubstrate material atom remain. In an embodiment, an etch rate maydepend on a density of plasma above a pattern element used as a maskingtemplate. In an embodiment, an etch rate may also depend on atemperature of substrate material and a kinetic energy of plasma speciesthat may accelerate to strike the substrate. For purposes of clarity,this application discusses embodiments involving a plasma etch (or areactive ion etch) process, although other means of generating devicefeatures are also envisioned.

Dimensions of pattern elements (corresponding to etched devicefeatures), or between pattern elements, of the device pattern may changeduring such a pattern transfer process and accordingly result in changein a dimension of the one or more associated device features. Inparticular, such change in dimensions may occur in a directionessentially parallel to the major plane of the substrate. In the contextof an etch process that involves removal of material, this may bereferred to as 2-dimensional (2D) etch bias. It is the lateral etchingof a device feature while a vertical dimension of the device feature isestablished, during the substrate material removal.

But, unanticipated or undesired etch bias may result in a device withperformance parameters outside of specification. For example, dimensionsof a device may be relevant to the function of the device. Variance indevice feature dimensions may modify electrical parameters of thedevice. Some parameters of an electronic device may be sensitive todevice dimensions including resistance and/or parasitic capacitance ofconductive lines, and/or timing of gate switching according to a gatelength of transistors in the device. Dielectric breakdown of aninsulating material in an integrated circuit may also be a function ofthe dimension of an insulating material. So, preserving a dimension of,or between, device features of a device may help ensure or preservefunctionality of the device within predetermined specifications.

Now, etch bias may vary on a substrate created by a patterning process.For example, etch bias may vary according to the density of a devicepattern being formed on a substrate. That is, etch bias may be differentin a dense region than in an isolated regions of a resist pattern.Additionally or alternatively, etch bias may be different at differentlocations of a substrate irrespective of density. For example, the etchbias may be different at a central portion of the substrate than at ornear an edge of the substrate.

Mathematically modeling etch bias may improve creation of the finaldevice feature dimensions. The results of such modeling can be used forvarious purposes. For example, the results can be used to adjust thepatterning process in terms of changing a design, controlling aparameter, etc. For example, the results can be used to adjust one ormore spatial properties of one or more elements in a pattern provided bythe patterning device, wherein the patterning device pattern is used tocreate the device pattern that will be used for etching on thesubstrate. Thus, once the adjusted patterning device pattern istransferred to the substrate, the device pattern on the substrate iseffectively adjusted, before etching, in order to compensate for etchbias that is expected to occur during etching. As another example, oneor more adjustments can be made to the lithographic apparatus in termsof adjustment of dose, focus, etc. As will be appreciated, there can bemany more applications. Thus, compensating for etch bias may result in adevice with more one or more uniform feature sizes, one or more uniformelectrical properties, and/or one or more improved (e.g., closer to adesired result) performance characteristics.

Moreover, etch bias, while sometimes detrimental to the manufacturing ofdevices on substrate, may sometimes be used in order to generatedesirable structures on a substrate. By accounting for a degree of etchbias when making a patterning device, it may be possible to manufacturedevice features in a device on the substrate with dimensions that aresmaller than the optical resolution limits of a pattern transfer processfrom the patterning device to the substrate. So, in this respect, suchmodeled results of etch bias can be used to adjust the patterningprocess in terms of changing a design, controlling a parameter, etc.

So, modeling etch bias in an etch process can help to generate moreaccurate device features by, for example, compensating for etch biassuch as by tailoring the patterning device to anticipate (correctly)what the etch bias of an etch process may be (e.g., as a function ofpattern density), so that the actual features generated by an etchprocess after (adjusted) lithography may be closer to a desired productspecification.

But, modeling etch bias can be a time-consuming, empirical task. It canalso be inaccurate. For example, modeling etch bias may be complex whenperformed at a full device pattern level (e.g., full chip level) becauseof the large number of features that are modeled Accordingly, there isprovided techniques here than quickly and accurately model etch bias toenable, for example, full chip analysis of patterning device patterns.

The present disclosure includes a description of a theoretical frameworkfor performing modeling of 2D etch bias that may be used in the processof making a patterning device for manufacturing a device. While thediscussion herein focuses on etch and etch bias, the principles of thetechniques can apply to other pattern transfer processes that transfer apattern in a layer into the substrate.

Etch bias may be measured on a substrate after an etch process isperformed. Etch bias may be a difference between a position of a portionof the side of a pattern element of a patterning layer (the patternelement corresponding to the device feature) prior to etch and theposition of the corresponding portion of the side of the pattern elementafter etch. Similarly, etch bias may be a difference between a positionof a portion of the side of a device feature (the device featurecorresponding to a pattern element) prior to etch and the position ofthe corresponding portion of the side of the device feature after etch.Etch bias may occur at an interface between an etched region of asubstrate and an unetched region of the substrate. In etch bias, anactual dimension or critical dimension (CD) of a device feature ordevice pattern element corresponding to the device feature may differfrom an anticipated or critical dimension of the device feature orpattern element (CD₀) after etch. In embodiment, the etch bias can bedefined at multiple locations on a device feature. A trench may haveetch bias at both sides of the trench, and a via may have etch bias atboth sides of the via. A line, such as for a finFET, may have etch biason both sides, making the final line dimension smaller than the patternfeature that masks the line during an etch process.

Etch bias may be sensitive to various conditions. For example, asdiscussed above, etch bias can be sensitive to local pattern density ina device pattern. Additionally or alternatively, etch bias may besensitive to chemistry conditions in the etching plasma, and/or to thetemperature of the substrate and/or plasma during the etching process.

FIG. 3 depicts a schematic diagram of an embodiment of a modeled area100 of a device pattern. Modeled area 100 may include device patternelements that correspond to etched device features that extend above amajor surface of a substrate and/or that are recessed into a substrate.Examples of recessed device features may include trenches, vias, and/orpad openings. Examples of protruding device features may include linesfor gates and/or finFETs, or masking lines compatible with doublepatterning lithography scenarios.

In an embodiment, pattern element 102 may be a trench in a patterninglayer (e.g., resist layer), or may be a line of patterning layer thatextends above a top surface of a substrate. While the discussion hereinwill focus on one or more pattern elements in a patterning layer, thediscussion herein could also be viewed in terms of the one or moreetched device features just as they are being formed as well as afterthey are formed.

Pattern element 102 may have a centerline 103 located between a firstside 104 and a second side 106 of the pattern element. In an embodimentwhere pattern element 102 is a trench, pattern element 102 may have afirst dimension 108 between first side 104 and second side 106, thedimension 108 corresponding to a dimension of a pattern element at thestart of an etch process to form one or more device features by etchingin the substrate using pattern element 102. Pattern element 102 may havea second dimension 110 between first side 104 and second side 106, thedimension 110 corresponding to a dimension of the pattern element afterthe one or more device features have been formed by etching using thepattern element 102. In an embodiment, if pattern element 102 is atrench, first dimension 108 may be smaller than second dimension 110.

According to an embodiment, first side 104 may have a first position104A at a beginning of an etch process and a second position 104B at theend of the etch process, where first position 104A and second position104B are different positions. Some characteristics of pattern element102 described above, as pertaining to a trench, may similarly beascribed to characteristics of a line that rises above a top surface ofa substrate.

Pattern element 102 may have a first etch bias 112A at first side 104and a second etch bias 112B at second side 106. The sum of first etchbias 112A and second etch bias 112B may be a dimension bias (or acritical dimension (CD) bias) of the pattern element that occurs duringformation of the one or more associated device features. First etch bias112A and second etch bias 112B may have different magnitudes or may besubstantially the same. A difference between first etch bias 112A andsecond etch bias 112B may relate to a density of pattern elements nearfirst side 104 and/or second side 106.

An etch bias may be positive, where a dimension of the pattern elementis larger after etching than before etching, or negative, where thedimension is smaller after etching than before etching. In anembodiment, a line formed by an etch process may have a negative etchbias after etching when an etch process recesses a sidewall of the linelaterally toward a central portion thereof during an etch process thatforms the line. In an embodiment a trench formed by an etch process mayhave a positive etch bias after etching when the etch process recesses asidewall of the line laterally away from a central portion of thetrench.

In an embodiment, pattern element 102 may have an initial area 114Ameasured between first side 104 and second 106 at first positions 104Aand 106A, respectively, and within a length 115 of the pattern element.In an embodiment, a final area 114B may be measured between secondpositions 104B and 106B, respectively. Initial area 114A may be smallerand/or larger than final area 114B according to an embodiment. Initialarea 114A and final area 114B may be etched areas of a device feature,describing a recessed area into, or protruding area from, a substratematerial.

Pattern element 102 may have an evaluation point 116 on a side of thepattern element. FIG. 3 shows evaluation point 116 on second side 106 atthe first position 106A. After a certain amount of etch time andassociated etch bias, the second side 106 may be located at evaluationpoint 124. So, the etch bias can be measured using measurements of oneor more spatial properties in relation to evaluation point 116/124. As asimple example, a measurement at a measurement site may involvemeasuring a position of evaluation point 116 can be measured before etchand a position of evaluation point 124 can be measured after etch. Asanother example, the cross-wise dimension of pattern element 102 may bemeasured before etch and after etch. As a further example, a dimension118 of a device pattern may be measured between pattern elements, suchas between pattern element 102 and an evaluation point 120 on a secondpattern element 122. So, an example measurement site in a device patternmay extend between evaluation points, such as evaluation point 116 andevaluation point 120. This dimension can be measured before etch andthen after etch. A device pattern on a substrate may have a plurality ofmeasurement sites (or locations) in the device pattern for measuringvalues of one or more spatial properties, including etch bias afteretching a substrate having a device pattern thereon. A measured devicepattern spatial property at a measurement site may be added to a set ofvalues of one or more spatial properties of a device pattern forcalculating etch bias. The etch bias calculation may be performed in acomputing device in an automated process as part of a manufacturingprocess or a product development process.

FIG. 4 depicts an example flow diagram of an embodiment of a method 200for determining etch bias. In an operation 202, a plurality of sites ina device pattern may be selected in order to measure etch bias in thedevice pattern. Sites in the plurality of sites may include patternelements corresponding to trenches, vias, pads, etc. In an embodiment, ameasurement site in the plurality of sites may traverse a patternelement such as a trench opening. In an embodiment, a measurement sitemay traverse material that lies between two pattern elements, such astwo trenches.

In an operation 204, a value of a first spatial property of the devicepattern may be measured at a plurality of measurement sites in a devicepattern. A first measurement of a device pattern may be performed aftera device pattern is transferred from a patterning device to a layer ofpatterning material on a substrate. A patterning material, such asresist, may be patterned by exposing the patterning material toradiation energy incident thereon from the patterning device. A doseand/or focus of the radiation may be adjusted in order to tailordimensions of the device pattern formed in the patterning material.Measurements may be recorded by optical, electrical, or other meanssuitable for probing one or more device pattern spatial properties on asubstrate according to processes and methods familiar to those in theart. In an embodiment, the set of values of the first spatial propertyat the measurement sites is input in to a storage medium or memory of acomputing device.

In an operation 206, substrate material (e.g., of the substrate baseand/or of a layer overlying the substrate base) may be removed from thesubstrate using the device pattern in the patterning layer as a maskingtemplate, e.g., by an etch process. The substrate material removal maybe a plasma etch process, a chemical etch process, or some othermaterial removal process that transfers a device pattern from the layerof patterning material on the substrate into substrate material.

In an operation 208, a value of second spatial property of the devicepattern may be measured after the substrate material removal process.Measurement of the second spatial property may be at a plurality ofmeasurement sites at or near the plurality of measurement sites wherethe first spatial property was measured. In an embodiment, the set ofvalues of the second spatial property at the measurement sites is inputin to a storage medium or memory of a computing device.

In an operation 210, an etch bias may be calculated for the measurementsites of the plurality of sites where first and second measurements wererecorded in operations 204 and 208, respectively. For example, adifference can be calculated between the applicable values of the firstand second properties. Etch bias values for the measurement sites of theplurality of sites may be recorded in a data set and optionally storedin a computer device memory for subsequent analysis and processing.Other information of the device pattern on the substrate may be recordedin the data set, including the location of measurement sites, a shape ofone or more pattern elements near to the measurement sites, and/or adimension of one or more pattern elements separate from the applicablepattern element that may be used for etch bias calculations, in order tofacilitate etch bias modeling.

In an operation 212, a mathematical model having a formula having one ormore mathematical terms with one or more variables and one or moreparameters is fitted to the etch bias, and/or other, data in the dataset. In an embodiment, fitting the data set with the formula results ina calculated value of one or more formula fitting parameters. Furtherdescription of the one or more variables and one or more parameters willdiscussed further hereafter.

In an operation 214, the parameterized model of operation 212 is used togenerate, for an etch process, one or more etch bias values for at leastone location in a device pattern. The results of such modeling can beused for various purposes. For example, the results can be used toadjust the patterning process in terms of changing a design, controllinga parameter, etc.

As an example of application of the determined etch bias, the resultscan be used to adjust one or more dimensions of one or more elements ina pattern provided by the patterning device, wherein the patterningdevice pattern is used to create the device pattern that will be usedfor etching on the substrate. Thus, once the adjusted patterning devicepattern is transferred to the substrate, the device pattern on thesubstrate is effectively adjusted, before etching, in order tocompensate for etch bias that is expected to occur during etching. Thatis, in an embodiment, an offset for a dimension of a patterning devicepattern feature is calculated based upon the determined etch bias tocompensate for etch bias in an etch process. In an embodiment, an offsetfor a dimension of a patterning device pattern feature is calculatedbased upon the determined etch bias in order to adjust a dimension of adevice pattern to match a feature dimension that is below an opticalresolution of the patterning device used to generate the device patternon a substrate.

As another example of application of the determined etch bias, one ormore adjustments can be made to the lithographic apparatus in terms ofadjustment of dose, focus, etc. As will be appreciated, there can bemany more applications. Thus, compensating for etch bias may result in adevice with more one or more uniform feature sizes, one or more uniformelectrical properties, and/or one or more improved (e.g., closer to adesired result) performance characteristics.

Formulas for making predictions of etch bias may take varioussophisticated forms. In an embodiment, an etch bias model may include aformula with a term related to a spatial property of a device patternprior to etching. In an embodiment, an etch bias model may include aformula with a term related to a spatial property of a device patternafter an etch process is performed. In an embodiment, an etch bias modelmay include a formula with a term that relates to an area of patterningmaterial surrounding a measuring site in the patterning device.

FIG. 5 depicts a cross-sectional diagram of an embodiment of a devicefeature and associated device pattern element 300 during an etchprocess. Plasma 302 is provided above substrate 304 comprising substratematerial 306 that is covered by a layer of patterning material 308. Thepatterning material has an opening 310 formed through, for example,resist development. The plasma 302 moves into the opening 310 andinteracts with substrate material 306 and the patterning material 308.Opening 310 enlarges downward in a first direction 314 by removal ofmaterial from surface 312, and expands laterally in a second direction318 through removal of material from side 316 and expands laterally in athird direction 322 through removal of material from side 320. The rateof material removal in the first direction 314 may exceed the rate ofetching in the second direction 318 and the third direction 322. Theopening 310 may have a first width 324 at the start of the etch process,and a second width 326 after the etch process ends. A volume 328 ofplasma 302 above the opening 310 may have a plasma density therein thatcan be used to model an etch process, or the etch bias resulting from anetch process.

A plasma surface concentration applied to a pattern element (trench)during an etch process may be represented by D with a unit such asmoles/μm². For purposes of embodiments of etch bias modeling describedherein, the plasma surface concentration may be approximated as aconstant value to simplify modeling calculations. A pattern element suchas opening 310 may have an area A with a unit such as (μm²) based on aninitial spatial property (such as an initial critical dimension CD₀)with a unit such as (μm) of the pattern element. In an embodiment, firstwidth 324 is an example of an initial spatial property of the patternelement. A number Q (moles) of atoms or molecules of a reactive speciesof a plasma above the initial area A may be determined as follows:

Q=D×A   [1]

A further approximation for the modeling process may include treatingthe number of reactive species in the trench as a constant, such as whenthe reactive species are in an equilibrium state, where the number ofincoming reactive species is equal to the number of exiting reactivespecies and the number of reactive species that are consumed by etchingand/or that bind to an etched surface. In other words, the rate ofchange of the number of reactive species is set to zero:

$\frac{dQ}{dt} = 0.$

Another approximation that may improve the ability to perform the etchbias modeling may involve treating the effect of reactive species in atrench as being similarly effective at lateral material removal at allpoints on a side of a pattern element. Thus, for a pattern elementperimeter L that extends around the area A, the plasma species linearconcentration (C_(T)) may be expressed as:

$\begin{matrix}{C_{T} = \frac{Q}{L}} & \lbrack 2\rbrack\end{matrix}$

and the lateral etch rate may be expressed as:

$\begin{matrix}{\frac{dCD}{dt} = {kC}_{T}^{n}} & \lbrack 3\rbrack\end{matrix}$

wherein k is a reaction constant, n is the reaction order, and CD is aspatial property (e.g., a dimension) of the pattern element (a trench inthis example). Where the pattern element is a recess, the spatialproperty may be a dimension that traverses an opening of the trench.Where the pattern element is not a recess, the spatial property may be adimension that traverses material of the pattern element that remainsas, or after, material is removed from the substrate.

For simple geometries such as circles, ellipses, and linear trenches,the plasma species linear concentration C_(T) may be defined as afunction of CD as follows:

$\begin{matrix}{{\ln \left( \frac{{CD}_{t}}{{CD}_{0}} \right)} = {kt}} & \lbrack 4\rbrack \\{{{CD}_{t} - {CD}_{0}} = {{{CD}_{0}e^{kt}} - {CD}_{0}}} & \lbrack 5\rbrack \\{{{etch}\mspace{14mu} {bias}} = {{CD}_{0}\left( {e^{kt} - 1} \right)}} & \lbrack 6\rbrack\end{matrix}$

wherein CD₀ represents the initial spatial property value (e.g., initialdimension), CD_(t) represents the spatial property at a later etch timet (and thus etch bias can be CD_(t)-CD₀), and k is a reaction constantfor the etch process.

Using an equation such as equation [6], the parameter kt can bedetermined by fitting the etch bias equation to a set of measurementdata. In particular, the parameter kt can be determined by fittingagainst a set of etch bias values determined for various values of CD₀based on data collected from a set of measurement sites (e.g., differentevaluation points 116/124) in a device pattern on a substrate asdescribed above. Thus, when parameterized by fitting, this etch biasmodel is specified for the particular etch process (including its etchtime) against which it is fitted.

So, in an embodiment, according to the model of equation [6], an etchbias of a device pattern etched using the etch process for which themodel of equation [6] has been parameterized can be determined by merelyinputting a particular value of CD₀. The etch bias model may be used fordifferent device patterns (using the same patterning process) and/or fordifferent locations of various different pattern elements of a devicepattern.

To accommodate a non-simple geometric layout (e.g., a random layout)where CD₀ is not readily defined, equation [6] can be reformulated as:

etch bias=k ₁ C _(T0)(e ^(kt)−1)   [7]

wherein t is the etch time, k is a reaction constant for the etchprocess, C_(T0) is the initial plasma species linear concentration, andk₁ is a calibration parameter to be fitted. This is done by recognizingthat CD₀ is approximately proportional to the initial plasma specieslinear concentration C_(T0) on the pattern element edge (note above thatC_(T) is effectively

$\left. \frac{D \times A}{L} \right).$

So, in embodiments where the CD₀ of a pattern element is unknown, or notwell defined, an approximation of CD₀ with C_(T0) may facilitate theetch bias calculation.

So, to determine C_(T0) for fitting of equation [7] and for subsequentdetermination of etch bias (since C_(T0) will be the variable in themodel effectively in place of CD₀ of the model of equation [6]), anambit can be defined that surrounds the evaluation point correspondingto which an etch bias has been measured (for purposes ofparameterization of the model) or corresponding to a location ofinterest for which an etch bias is desired (for purpose of etch biascalculation using the parameterized model). The ambit effectivelydelimits the initial plasma specifies linear concentration.

FIG. 6 depicts a schematic diagram of a modeled area 400 of anembodiment of a device. Modeled area 400 includes a pattern element 402(a trench) with an area 404 located within an initial area perimeter405. Pattern element 402 has an evaluation point 410. Evaluation 410corresponds to the location where an etch bias is determined forparameterization of the model and/or corresponds to the location forwhich is an etch bias is determined using a parameterized mode. So, todefine the initial plasma species linear concentration, an ambit 406 isdefined. In an embodiment, the ambit 406 is a radius. In an embodiment,the ambit 406 is defined relative to the evaluation point 410, e.g.,evaluation point 410 is at a central portion of the ambit 406. Ambit 406helps define the extent of the area and perimeter of pattern element 402(since evaluation point 410 is located on pattern element 402) that willbe considered to calculate C_(T0), which is effectively

$\frac{D \times A}{L}.$

That is, it helps define the area adjoining evaluation point 410. So, asseen in FIG. 6, the applicable area is the shaded area 404, which asseen is delimited by an ambit 406 at opposite ends thereof and delimitedby opposite sides of the pattern element 402. As seen in FIG. 6, ambit408 extends to overlap a second pattern element 416, but that area isnot included because it is not connected to pattern element (trench)402. So, the initial plasma species linear concentration C_(T0) can becalculated using the area and perimeter of the shaded area 404 incombination with the plasma specifies surface concentration in theformula

$\frac{D \times A}{L}.$

For the purpose of parameterization of the model, initial plasma specieslinear concentration C_(T0) would be calculated for each etch biasevaluated for fitting. For the purpose of etch bias calculation using aparameterized the model, initial plasma species linear concentrationC_(T0) would be calculated for the evaluation point 410 of interest.

So, using an equation such as equation [7], the parameters kt and k₁ canbe determined by fitting the etch bias equation to a set of measurementdata. In particular, the parameters kt and k₁ can be determined byfitting against a set of etch bias values determined for various initialplasma species linear concentrations C_(T0) calculated as describedabove, collected from a set of measurement sites (e.g., differentevaluation points 410) in a device pattern on a substrate as describedabove. Thus, when parameterized by fitting, this etch bias model isspecified for the particular etch process against which it is fittedincluding the etch time and ambit. The ambit size describes how long theetch proximity effects are and so can be decided based on substrate data(e.g., different ambit sizes can be tried to get better fitting with ahigher or highest coefficient of determination).

So, in an embodiment, according to the model of equation [7], an etchbias of a device pattern etched using the etch process for which themodel of equation [7] has been parameterized can be determined by merelyinputting a particular initial plasma species linear concentrationC_(T0) (which can be calculated as described using the ambit size of themodel).

While the etch bias calculations above have focused on the etch biascontributions from plasma species themselves, the patterning material(e.g., resist or other masking material) itself can affect the nature ofthe etch bias. Thus, an etch bias calculation may factor in one or morematerials of the patterning material that impact the etch bias of thesubstrate. So, the etch bias contribution of a patterning materialitself can be factored in by relating the reaction constant k with oneor more patterning material characteristics relating to the etch rate inthe lateral direction. So, in an embodiment, the rate constant k ofequation [7] may be replaced with a modified form of the Arrheniusequation. By substituting a linear concentration of patterning materialC_(R) for the activation energy Ea of the Arrhenius equation andcombining the other factors of the exponent into a term s, a modifiedform of the Arrhenius equation may be expressed as:

$\begin{matrix}{k = {Ae}^{\frac{- C_{R}}{s}}} & \lbrack 8\rbrack\end{matrix}$

and the etch bias equation [7] may then be expressed as follows:

$\begin{matrix}{{{etch}\mspace{14mu} {bias}} = {k_{1}{C_{T\; 0}\left( {e^{{Ate}^{\frac{- C_{R}}{s}}} - 1} \right)}}} & \lbrack 9\rbrack\end{matrix}$

wherein t is the etch time, C_(T0) is the initial plasma species linearconcentration (which can be calculated as described above), k₁ is acalibration parameter to be fitted, and A is the frequency factor forthe reaction.

While C_(T0) relates to the exposed initial etch area of the patternelement being etched, patterning material linear concentration C_(R)relates to the perimeter area of the patterning material that adjoinsthe particular evaluation point of the device pattern. That is, an areaof patterning material can be specified to determine the patterningmaterial linear concentration C_(R) corresponding to a point on theperimeter of the pattern element that is being etched. The followingfigures demonstrate different example techniques to provide a linearconcentration of patterning material for the etch bias formula shownabove in equation [9].

FIG. 7 depicts a schematic diagram of a modeled area 500 of anembodiment of a device. Modeled area 500 includes a pattern element 502.In this case, pattern element 502 is a trench. In other embodiments,pattern element 502 may be a different type of feature. Modeled area 500includes an etched area 504 within the sides 506A and 506B of patternelement 502 and within segments 508A and 508B of an ambit 508 locatedaround an evaluation point 510 of the modeled area 502. Ambit 508 may bea circular shape, as shown in FIG. 7, or may be another closed shapesuch as an ellipse, an oval, a rectilinear shape, or some other shapethat encloses a portion of a pattern element and a region of patterningmaterial adjoining the etched area 504 of the pattern element 502. InFIG. 7, ambit 508 has the evaluation point 510 at a central portion ofthe ambit 508, with a radius 512 extending outward from evaluation point510 to describe an ambit perimeter.

FIG. 7 shows a patterning material (e.g. resist) area 516 within ambit508 that may be bounded by a side 506B of the pattern element 502 andperimeter segments 514A and 514B of the ambit 508. Patterning materialarea 516 may exclude an open area 520 that is within a border 522 of aneighboring pattern element 524 that falls at least partly within theperimeter of the ambit 508. In some embodiments, an ambit 508 mayinclude a plurality of open areas such as open area 520 that are locatedfully or partly within the perimeter of the ambit 508 and reduce thesize of the patterning material area 516 that may be used to calculatethe patterning material linear concentration for the patterning materialadjoining the evaluation point and thus affecting the etch bias at theevaluation point.

FIG. 8 depicts a schematic diagram of a modeled area 600 of anembodiment of a device. Modeled area 600 may include a first patternelement 602 with a first sidewall 604, and a second pattern element 606with a second sidewall 608. An evaluation point 610 is at one end offirst pattern element 602. Evaluation point 610 lies at one side ofpatterning material ambit 614 that extends toward second pattern element606. Ambit 614 extends a first distance 620 outward from first patternelement 602 and a second distance 622 perpendicular to first distance622. In an embodiment, the first distance 820 is perpendicular to a sideor tangent of the sidewall 604. Ambit 614 may encompass patterningmaterial area 616 and overlap area 618. The overlap area 618 correspondsto the portion of the ambit 614 extending across at least part ofanother pattern element, such as pattern element 606. So, the patterningmaterial area 616 can be defined as the ambit less the overlap area 618.Ambit 614 may be a rectilinear ambit but could be circular, oval,elliptical, or some other closed shape that encompasses or adjoins atleast an evaluation point, such as evaluation point 610 of patternelement 602. In some embodiments, the ambit 614 can be filled entirelywith patterning material if the ambit 614 were to extend and not includean overlap area such as overlap area 618 (e.g., if ambit 614 extendedfrom an appropriate different evaluation point on sidewall 604).

So, for area 516 or 616, the patterning material linear concentrationC_(R) can be calculated. It can be calculated using a similarformulation as C_(T) described above except using a pattern materialsurface concentration instead of plasma species surface concentration.

Then, using equation [9], the parameters At and k₁ can be determined byfitting the etch bias equation to a set of measurement data. Inparticular, the parameters At and k₁ can be determined by fittingagainst a set of etch bias values determined for various initial plasmaspecies linear concentrations C_(T0) calculated as described above andassociated patterning material linear concentrations C_(R) calculated asdescribed above, collected from a set of measurement sites (e.g.,different evaluation points 410, 510, 610) in a device pattern on asubstrate as described above. Thus, when parameterized by fitting, thisetch bias model is specified for the particular etch process againstwhich it is fitted including the etch time and associated ambits. Theambits can be decided based on substrate data (e.g., different ambitsizes and/or shapes can be tried to get better fitting with a higher orhighest coefficient of determination).

So, in an embodiment, according to the model of equation [9], an etchbias of a device pattern etched using the etch process for which themodel of equation [9] has been parameterized can be determined by merelyinputting a particular initial plasma species linear concentrationC_(T0) (which can be calculated as described using the ambit size of themodel) and patterning material linear concentrations C_(R) (which can becalculated as described using the ambit size of the model).

In the above formulations, the patterning material layer materialsproperty is related to the reaction constant. But, it may be that etchbias is a chemical reaction that occurs with the plasma species and thepatterning material participating as reactants. Then, etch bias in thelateral direction can be treated as a second order reaction, with thelateral etch rate being proportional to both C_(T) and C_(R).

$\begin{matrix}{\frac{dE}{dt} = {{kC}_{T}C_{R}}} & \lbrack 10\rbrack\end{matrix}$

wherein k is a reaction constant.

It can be approximately assumed that the decrease of C_(T) equals theincrease of C_(R), or vice versa. So, at time t, the decrease of C_(T)(and the increase of C_(R)) can be designated as x, then it can beformulated that:

$\begin{matrix}{\frac{dx}{dt} = {{k\left( {C_{T\; 0} - x} \right)}\left( {C_{R\; 0} + x} \right)}} & \lbrack 11\rbrack \\{\frac{dx}{\left( {C_{T\; 0} - x} \right)\left( {C_{R\; 0} + x} \right)} = {kdt}} & \lbrack 12\rbrack\end{matrix}$

wherein C_(T0) is C_(T) at time 0 and C_(R0) is C_(R) at time 0. So,using an initial condition of x=0 at t=0, and integrating the abovedifferential equation yields:

$\begin{matrix}{{\frac{1}{{- C_{R\; 0}} - C_{T\; 0}}{\ln \left( \frac{C_{T\; 0}\left( {C_{R\; 0} + x} \right)}{C_{R\; 0}\left( {C_{T\; 0} - x} \right)} \right)}} = {kt}} & \lbrack 13\rbrack \\{x = {C_{T\; 0} - \frac{C_{T\; 0} - C_{R\; 0}}{1 + {\frac{C_{R\; 0}}{C_{T\; 0}}e^{- {{kt}{({C_{T\; 0} + C_{R\; 0}})}}}}}}} & \lbrack 14\rbrack \\{{C_{T}(t)} = {{C_{T\; 0} - x} = \frac{C_{T\; 0} + C_{R\; 0}}{1 + {\frac{C_{R\; 0}}{C_{T\; 0}}e^{- {{kt}{({C_{T\; 0} + C_{R\; 0}})}}}}}}} & \lbrack 15\rbrack \\{{C_{R}(t)} = {{C_{R\; 0} + x} = {\left( {C_{T\; 0} + C_{R\; 0}} \right)\left( {1 - \frac{1}{1 + {\frac{C_{R\; 0}}{C_{T\; 0}}e^{- {{kt}{({C_{T\; 0} + C_{R\; 0}})}}}}}} \right)}}} & \lbrack 16\rbrack\end{matrix}$

wherein C_(T)(t) is C_(T) at time t and C_(R)(t) is C_(R) at time t.Since, in the two equations above, C_(T) and C_(R) are expressed asfunctions of etch time, the final etch bias can be expressed as:

bias=Σ₀ ^(n) c _(n) C _(T)(t _(n))C _(R)(t _(n))   [17]

wherein C_(T) and C_(R) are evaluated at n time intervals in the etchtime, and t_(n) and c_(n) are etch rate coefficients. With measurementdata, the etch rate coefficients c_(n) along with other parameters forevaluating C_(T) and C_(R) can be calibrated to have an etch bias model.

Thus, there is provided an etch bias model based on simplified chemicaldynamics that aims at predicting/simulating lateral CD evolvement due topattern proximity effects. Etch bias is ascribed to plasma species in atrench and optionally etched patterning material in the neighborhood.

A concept in the model is that the plasma species uniformly act on theedges to induce an etch bias. An assumption that the plasma speciesmaintain an equilibration state may not be true due to the oscillatorynature of the strong RF (radio frequency) electromagnetic field used toinitiate plasma. The failure of such an assumption may lead toinaccurate etch time estimation. But, with focus on a lateral spatialproperty (e.g., CD) change and the time factor in the model is a fittingparameter, the failure of such an assumption does not significantlyaffect the effectiveness of the model.

Regarding etch bias contribution from etched patterning material,several approaches have been discussed above. The contribution of thepatterning material to etch bias can be treated as an exponential factorin the reaction constant just like the classical Arrhenius equationgoverning reactions. Additionally or alternatively, the patterningmaterial can be treated as one reactant in a second order reactionscheme.

So, a physical approach to modelling etch bias for etch processes hasbeen described herein. The approach is capable of simulating etch biasfor various layouts with potential full-chip applications. In anembodiment, it assumes that plasma chemical species in the trenches aremaintained in an equilibrium state and the plasma species act on edgesuniformly to induce etch bias. For complex layouts, methods wereprovided to evaluate edge loadings of plasma species. This evaluation isbased on local neighboring trench area and edge length. In addition, theimpact of patterning material on etch bias can be incorporated in anetch bias model in several ways. One way to do so is to treat the impactas an exponential factor in the reaction constant. A further way treatsthe patterning material as a reactant (together with plasma species) ina second order reaction scheme wherein the evolvement of C_(T) and C_(R)as a function of time is derived and the etch bias is a time integral ofC_(T) and C_(R) over time. A set of calibration data are used tocalibrate the etch bias model with this physical approach.

In an embodiment, there is provided a method, comprising: determining,by a hardware computer, an etch bias for a pattern to be etched using anetch step of a patterning process based on an etch bias model, the etchbias model comprising a formula including a variable associated with aspatial property of the pattern or with an etch plasma speciesconcentration of the etch step, and including a mathematical termcomprising a natural exponential function to the power of a parameterthat is fitted or based on an etch time of the etch step; and adjustingthe patterning process based on the determined etch bias. In anembodiment, the parameter of the exponential function is fitted or basedon the etch time and a reaction constant for the etch step. In anembodiment, the variable comprises the spatial property of the patternand the spatial property of the pattern is an initial pattern elementdimension. In an embodiment, n the variable comprises the spatialproperty of the pattern and the formula comprises the variablemultiplied by the mathematical term. In an embodiment, the variablecomprises the spatial property of the pattern and the formula comprisesa form of CD₀(e^(kt)−1), where CD₀ is the variable and corresponds to adimension of the pattern and kt is the parameter that is fitted for theetch time t of the etch step and a reaction constant k for the etchstep. In an embodiment, the variable comprises the etch plasma speciesconcentration and the formula further comprises a calibration parameter.In an embodiment, the variable comprises the etch plasma speciesconcentration and the formula comprises the variable multiplied by themathematical term. In an embodiment, the variable comprises the etchplasma species concentration wherein the formula comprises a form ofk₁C_(T0)(e^(kt)−1), wherein k₁ is a calibration parameter, C_(T0) is thevariable and corresponds to the etch plasma species concentration, andkt is the parameter that is fitted for the etch time t of the etch stepand a reaction constant k for the etch step. In an embodiment, thevariable comprises the etch plasma species concentration and the etchplasma species concentration is defined for an etched area of thepattern within an etched material ambit surrounding an evaluation pointon the pattern, wherein the etch plasma species concentration isproportional to the etched area. In an embodiment, the etched materialambit is a circular ambit centered on the evaluation point, theevaluation point being located at an interface between the etched areaand a patterning material area of the substrate. In an embodiment, thevariable comprises the etch plasma species concentration and the formulacontains a modified form of the Arrhenius equation incorporated in thepower of the exponential function. In an embodiment, the variablecomprises the etch plasma species concentration and the formulaincorporates a patterning material concentration in the power of theexponential function. In an embodiment, the variable comprises the etchplasma species concentration and wherein the formula has a formcomprising

${k_{1}{C_{T\; 0}\left( {e^{{Ate}^{\frac{- {CR}}{s}}} - 1} \right)}},$

where k₁ is a calibration parameter, C_(T0) is the variable andcorresponds to the etch plasma species concentration, C_(R) is apatterning material concentration, At is the parameter that is fittedfor the etch time t of the etch step and a frequency factor A for thereaction of the etch step, and s is a constant for the etch step. In anembodiment, the patterning material concentration is defined for apatterning material area of the pattern adjacent an evaluation point onthe pattern. In an embodiment, the patterning material ambit isrectilinear and the patterning material ambit adjoins or overlays theevaluation point or is circular and surrounds the evaluation point. Inan embodiment, the etch plasma species concentration is defined for anetched area of the pattern within an etched material ambit surroundingan evaluation point on the pattern, wherein the etch plasma speciesconcentration is proportional to the etched area. In an embodiment, themethod further comprises: collecting, at each of a plurality of sites ina pattern, a value of a spatial property of the pattern; and fitting, bya hardware computing device and using the values of the spatialproperty, the formula to generate the parameter. In an embodiment,adjusting the patterning process comprises adjusting a border of aregion of the patterning device according to the calculated etch bias.In an embodiment, the region of the patterning device modifies radiationthat strikes the patterning device. In an embodiment, the pattern is adevice pattern.

In an embodiment, there is provided a method, comprising: determining,by a hardware computer, an etch bias for a pattern to be etched using anetch step of a patterning process based on an etch bias model, the etchbias model comprising a function of an etch plasma species concentrationand a patterning material concentration; and adjusting the patterningprocess based on the determined etch bias.

In an embodiment, the function comprises the etch plasma speciesconcentration multiplied with the patterning material concentration. Inan embodiment, the function comprises a summation of the etch plasmaspecies concentration and the patterning material concentration for acertain number of time intervals of an etch time. In an embodiment, theetch model comprises a mathematical term comprising a naturalexponential function to the power of a parameter that is fitted or basedon an etch time of the etch step. In an embodiment, the etch bias modelhas a form comprising Σ₀ ^(n)c_(n)C_(T)(t_(n))C_(R)(t_(n)), whereinC_(T) corresponds to the etch plasma species concentration and isevaluated at n number of time intervals in the etch time, C_(R)corresponds to the patterning material concentration and is evaluated atn number of time intervals in the etch time, and t_(n) and c_(n) areetch rate coefficients.

As will be appreciated by one of ordinary skill in the art, the presenttechniques may be embodied as a system, method, or computer programproduct. Accordingly, aspects of the present application may take theform of an entirely hardware embodiment, an entirely software embodiment(including firmware, resident software, micro-code, etc.) or anembodiment combining software and hardware aspects that may allgenerally be referred to herein as a “circuit,” “module” or “system.”Furthermore, aspects of the present application may take the form of acomputer program product embodied in any one or more computer readablemedium(s) having computer usable program code embodied thereon.

Any combination of one or more computer readable medium(s) may beutilized. The computer readable medium may be a computer readable signalmedium or a computer readable storage medium. A computer readablestorage medium may be, for example, but not limited to, an electronic,magnetic, optical, electromagnetic, infrared, or semiconductor system,apparatus, device, or any suitable combination of the foregoing. Morespecific examples (a non-exhaustive list) of the computer readablemedium would include the following: an electrical connection having oneor more wires, a portable computer diskette, a hard disk, a randomaccess memory (RAM), a read-only memory (ROM), an erasable programmableread-only memory (e.g. EPROM or Flash memory), an optical fiber, aportable compact disc read-only memory CDROM, an optical storage device,a magnetic storage device, or any suitable combination of the foregoing.In the context of this document, a computer readable storage medium maybe any tangible medium that can contain or store a program for use by orin connection with an instruction execution system, apparatus, ordevice.

A computer readable signal medium may include a propagated data signalwith computer readable program code embodied therein, for example, in abaseband or as part of a carrier wave. Such a propagated signal may takeany of a variety of forms, including, but not limited to,electro-magnetic, optical, or any suitable combination thereof. Acomputer readable signal medium may be any computer readable medium thatis not a computer readable storage medium and that can communicate,propagate, or transport a program for use by or in connection with aninstruction execution system, apparatus, or device.

Computer code embodied on a computer readable medium may be transmittedusing any appropriate medium, including but not limited to wireless,wireline, optical fiber cable, radio frequency RF, etc., or any suitablecombination thereof.

Computer program code for carrying out operations for aspects of thepresent application may be written in any combination of one or moreprogramming languages, including an object oriented programming languagesuch as Java™, Smalltalk™, C++, or the like, and conventional proceduralprogramming languages, such as the “C” programming language or similarprogramming languages. The program code may execute entirely on theuser's computer, partly on the user's computer, as a stand-alonesoftware package, partly on the user's computer and partly on a remotecomputer, or entirely on the remote computer or server. In the latterscenario, the remote computer may be connected to the user's computerthrough any type of network, including a local area network LAN or awide area network WAN, or the connection may be made to an externalcomputer (for example, through the Internet using an Internet ServiceProvider).

The computer program instructions may also be loaded onto a computer,other programmable data processing apparatus, or other devices to causea series of operational steps to be performed on the computer, otherprogrammable apparatus, or other devices to produce a computerimplemented process such that the instructions which execute on thecomputer or other programmable apparatus provide processes forimplementing one or more of the functions/acts specified herein.

As noted above, it should be appreciated that the illustrativeembodiments may take the form of an entirely hardware embodiment, anentirely software embodiment or an embodiment containing both hardwareand software elements. In one example embodiment, the mechanisms of theillustrative embodiments may be implemented in software or program code,which includes but is not limited to firmware, resident software,microcode, etc.

A data processing system suitable for storing and/or executing programcode will include at least one processor coupled directly or indirectlyto memory elements through a system bus. The memory elements can includelocal memory employed during actual execution of the program code, bulkstorage, and cache memories which provide temporary storage of at leastsome program code in order to reduce the number of times code must beretrieved from bulk storage during execution.

Input/output or I/O devices (including but not limited to keyboards,displays, pointing devices, etc.) can be coupled to the system eitherdirectly or through intervening I/O controllers. Network adapters mayalso be coupled to the system to enable the data processing system tobecome coupled to other data processing systems or remote printers orstorage devices through intervening private or public networks. Modems,cable modems and Ethernet cards are just a few of the currentlyavailable types of network adapters.

FIG. 9 shows a block diagram that illustrates an embodiment of acomputer system 1700 which can assist in implementing any of the methodsand flows disclosed herein. Computer system 1700 includes a bus 1702 orother communication mechanism for communicating information, and aprocessor 1704 (or multiple processors 1704 and 1705) coupled with bus1702 for processing information. Computer system 1700 also includes amain memory 1706, such as a random access memory RAM or other dynamicstorage device, coupled to bus 1702 for storing information andinstructions to be executed by processor 1704. Main memory 1806 also maybe used for storing temporary variables or other intermediateinformation during execution of instructions to be executed by processor1704. Computer system 1700 further includes a read only memory ROM 1708or other static storage device coupled to bus 1702 for storing staticinformation and instructions for processor 1704. A storage device 1710,such as a magnetic disk or optical disk, is provided and coupled to bus1702 for storing information and instructions.

Computer system 1700 may be coupled via bus 1702 to a display 1712, suchas a cathode ray tube (CRT) or flat panel or touch panel display fordisplaying information to a computer user. An input device 1714,including alphanumeric and other keys, is coupled to bus 1702 forcommunicating information and command selections to processor 1704.Another type of user input device is cursor control 1716, such as amouse, a trackball, or cursor direction keys for communicating directioninformation and command selections to processor 1704 and for controllingcursor movement on display 1712. This input device typically has twodegrees of freedom in two axes, a first axis (e.g. x) and a second axis(e.g. y), that allows the device to specify positions in a plane. Atouch panel (screen) display may also be used as an input device.

According to one embodiment, portions of a process described herein maybe performed by computer system 1700 in response to processor 1704executing one or more sequences of one or more instructions contained inmain memory 1706. Such instructions may be read into main memory 1706from another computer-readable medium, such as storage device 1710.Execution of the sequences of instructions contained in main memory 1706causes processor 1704 to perform the process steps described herein. Oneor more processors in a multi-processing arrangement may also beemployed to execute the sequences of instructions contained in mainmemory 1706. In an alternative embodiment, hard-wired circuitry may beused in place of or in combination with software instructions. Thus, thedescription herein is not limited to any specific combination ofhardware circuitry and software.

The term “computer-readable medium” as used herein refers to any mediumthat participates in providing instructions to processor 1704 forexecution. Such a medium may take many forms, including but not limitedto, non-volatile media, volatile media, and transmission media.Non-volatile media include, for example, optical or magnetic disks, suchas storage device 1710. Volatile media include dynamic memory, such asmain memory 1706. Transmission media include coaxial cables, copper wireand fiber optics, including the wires that comprise bus 1702.Transmission media can also take the form of acoustic or light waves,such as those generated during radio frequency (RF) and infrared (IR)data communications. Common forms of computer-readable media include,for example, a floppy disk, a flexible disk, hard disk, magnetic tape,any other magnetic medium, a CD-ROM, DVD, any other optical medium,punch cards, paper tape, any other physical medium with patterns ofholes, a RAM, a PROM, and EPROM, a FLASH-EPROM, any other memory chip orcartridge, a carrier wave as described hereinafter, or any other mediumfrom which a computer can read.

Various forms of computer readable media may be involved in carrying oneor more sequences of one or more instructions to processor 1704 forexecution. For example, the instructions may initially be borne on amagnetic disk of a remote computer. The remote computer can load theinstructions into its dynamic memory and send the instructions over atelephone line using a modem. A modem local to computer system 1700 canreceive the data on the telephone line and use an infrared transmitterto convert the data to an infrared signal. An infrared detector coupledto bus 1702 can receive the data carried in the infrared signal andplace the data on bus 1702. Bus 1702 carries the data to main memory1706, from which processor 1704 retrieves and executes the instructions.The instructions received by main memory 1706 may optionally be storedon storage device 1710 either before or after execution by processor1704.

Computer system 1700 may also include a communication interface 1718coupled to bus 1702. Communication interface 1718 provides a two-waydata communication coupling to a network link 1720 that is connected toa local network 1722. For example, communication interface 1718 may bean integrated services digital network ISDN card or a modem to provide adata communication connection to a corresponding type of telephone line.As another example, communication interface 1718 may be a local areanetwork LAN card to provide a data communication connection to acompatible LAN. Wireless links may also be implemented. In any suchimplementation, communication interface 1718 sends and receiveselectrical, electromagnetic or optical signals that carry digital datastreams representing various types of information.

Network link 1720 typically provides data communication through one ormore networks to other data devices. For example, network link 1720 mayprovide a connection through local network 1722 to a host computer 1724or to data equipment operated by an Internet Service Provider ISP 1726.ISP 1726 in turn provides data communication services through theworldwide packet data communication network, now commonly referred to asthe “Internet” 1728. Local network 1722 and Internet 1728 both useelectrical, electromagnetic or optical signals that carry digital datastreams. The signals through the various networks and the signals onnetwork link 1720 and through communication interface 1718, which carrythe digital data to and from computer system 1700, are exemplary formsof carrier waves transporting the information.

Computer system 1700 can send messages and receive data, includingprogram code, through the network(s), network link 1720, andcommunication interface 1718. In the Internet example, a server 1730might transmit a requested code for an application program throughInternet 1728, ISP 1726, local network 1722 and communication interface1718. One such downloaded application may provide for a method orportion thereof as described herein, for example. The received code maybe executed by processor 1704 as it is received, and/or stored instorage device 1710, or other non-volatile storage for later execution.In this manner, computer system 1700 may obtain application code in theform of a carrier wave.

The embodiments may further be described using the following clauses:

-   1. A method, comprising:

determining, by a hardware computer, an etch bias for a pattern to beetched using an etch step of a patterning process based on an etch biasmodel, the etch bias model comprising a formula including a variableassociated with a spatial property of the pattern or with an etch plasmaspecies concentration of the etch step, and including a mathematicalterm comprising a natural exponential function to the power of aparameter that is fitted or based on an etch time of the etch step; and

adjusting the patterning process based on the determined etch bias.

-   2. The method of clause 1, wherein the parameter of the exponential    function is fitted or based on the etch time and a reaction constant    for the etch step.

3. The method of clause 1 or clause 2, wherein the variable comprisesthe spatial property of the pattern and the spatial property of thepattern is an initial pattern element dimension.

-   4. The method of any of clauses 1-3, wherein the variable comprises    the spatial property of the pattern and the formula comprises the    variable multiplied by the mathematical term.-   5. The method of any of clauses 1-4, wherein the variable comprises    the spatial property of the pattern and the formula comprises a form    of CD₀(e^(kt)−1), where CD₀ is the variable and corresponds to a    dimension of the pattern and kt is the parameter that is fitted for    the etch time t of the etch step and a reaction constant k for the    etch step.-   6. The method of any of clauses 1-5, wherein the variable comprises    the etch plasma species concentration and the formula further    comprises a calibration parameter.-   7. The method of any of clauses 1-6, wherein the variable comprises    the etch plasma species concentration and the formula comprises the    variable multiplied by the mathematical term.-   8. The method of any of clauses 1-7, wherein the variable comprises    the etch plasma species concentration wherein the formula comprises    a form of k₁C_(T0)(e ^(kt)−1), wherein k₁ is a calibration    parameter, C_(T0) is the variable and corresponds to the etch plasma    species concentration, and kt is the parameter that is fitted for    the etch time t of the etch step and a reaction constant k for the    etch step.-   9. The method of any of clauses 1-8, wherein the variable comprises    the etch plasma species concentration and the etch plasma species    concentration is defined for an etched area of the pattern within an    etched material ambit surrounding an evaluation point on the    pattern, wherein the etch plasma species concentration is    proportional to the etched area.-   10. The method of clause 9, wherein the etched material ambit is a    circular ambit centered on the evaluation point, the evaluation    point being located at an interface between the etched area and a    patterning material area of the substrate.-   11. The method of any of clauses 1-10, wherein the variable    comprises the etch plasma species concentration and the formula    contains a modified form of the Arrhenius equation incorporated in    the power of the exponential function.-   12. The method of any of clauses 1-11, wherein the variable    comprises the etch plasma species concentration and the formula    incorporates a patterning material concentration in the power of the    exponential function.-   13. The method of any of clauses 1-12, wherein the variable    comprises the etch plasma species concentration and wherein the    formula has a form comprising

${k_{1}{C_{T\; 0}\left( {e^{{Ate}^{\frac{- {CR}}{s}}} - 1} \right)}},$

where k₁ is a calibration parameter, C_(T0) is the variable andcorresponds to the etch plasma species concentration, C_(R) is apatterning material concentration, At is the parameter that is fittedfor the etch time t of the etch step and a frequency factor A for thereaction of the etch step, and s is a constant for the etch step.

-   14. The method of clause 12 or clause 13, wherein the patterning    material concentration is defined for a patterning material area of    the pattern adjacent an evaluation point on the pattern.-   15. The method of clause 14, wherein the patterning material ambit    is rectilinear and the patterning material ambit adjoins or overlays    the evaluation point or is circular and surrounds the evaluation    point.-   16. The method of any of clauses 12-15, wherein the etch plasma    species concentration is defined for an etched area of the pattern    within an etched material ambit surrounding an evaluation point on    the pattern, wherein the etch plasma species concentration is    proportional to the etched area.-   17. The method of any of clauses 1-16, further comprising:

collecting, at each of a plurality of sites in a pattern, a value of aspatial property of the pattern; and

fitting, by a hardware computing device and using the values of thespatial property, the formula to generate the parameter.

-   18. The method of any of clauses 1-17, wherein adjusting the    patterning process comprises adjusting a border of a region of the    patterning device according to the calculated etch bias.-   19. The method of clause 18, wherein the region of the patterning    device modifies radiation that strikes the patterning device.-   20. The method of any of clauses 1-19, wherein the pattern is a    device pattern.-   21. A method, comprising:

determining, by a hardware computer, an etch bias for a pattern to beetched using an etch step of a patterning process based on an etch biasmodel, the etch bias model comprising a function of an etch plasmaspecies concentration and a patterning material concentration; and

adjusting the patterning process based on the determined etch bias.

-   22. The method of clause 21, wherein the function comprises the etch    plasma species concentration multiplied with the patterning material    concentration.-   23. The method of clause 21 or clause 22, wherein the function    comprises a summation of the etch plasma species concentration and    the patterning material concentration for a certain number of time    intervals of an etch time.-   24. The method of any of clauses 21-23, wherein the etch model    comprises a mathematical term comprising a natural exponential    function to the power of a parameter that is fitted or based on an    etch time of the etch step.-   25. The method of any of clauses 21-24, wherein the etch bias model    has a form comprising Σ₀ ^(n)c_(n)C_(T)(t_(n))C_(R)(t_(n)), wherein    C_(T) corresponds to the etch plasma species concentration and is    evaluated at n number of time intervals in the etch time, C_(R)    corresponds to the patterning material concentration and is    evaluated at n number of time intervals in the etch time, and t_(n)    and c_(n) are etch rate coefficients.-   26. A computer program product comprising a non-transitory computer    readable medium having instructions recorded thereon, the    instructions when executed by a computer implementing the method of    any of clauses 1-25.

Although specific reference may be made in this text to the manufactureof ICs, it should be explicitly understood that the description hereinhas many other possible applications. For example, it may be employed inthe manufacture of integrated optical systems, guidance and detectionpatterns for magnetic domain memories, liquid crystal display panels,thin film magnetic heads, etc. The skilled artisan will appreciate that,in the context of such alternative applications, any use of the terms“reticle”/“mask”, “wafer” or “die” in this text should be considered asinterchangeable with the more general terms “patterning device”,“substrate” and “target portion”, respectively.

In the present document, the terms “radiation” and “beam” are used toencompass all types of electromagnetic radiation, including ultravioletradiation (e.g. with a wavelength of 365, 248, 193, 157 or 126 nm) andEUV (extreme ultra-violet radiation, e.g. having a wavelength in therange of about 5-100 nm).

While the concepts disclosed herein may be used with systems and methodsfor imaging on a substrate such as a silicon wafer, it shall beunderstood that the disclosed concepts may be used with any type oflithographic systems, e.g., those used for imaging on substrates otherthan silicon wafers.

In block diagrams, illustrated components are depicted as discretefunctional blocks, but embodiments are not limited to systems in whichthe functionality described herein is organized as illustrated. Thefunctionality provided by each of the components may be provided bysoftware or hardware modules that are differently organized than ispresently depicted, for example such software or hardware may beintermingled, conjoined, replicated, broken up, distributed (e.g. withina data center or geographically), or otherwise differently organized Thefunctionality described herein may be provided by one or more processorsof one or more computers executing code stored on a tangible,non-transitory, machine readable medium. In some cases, third partycontent delivery networks may host some or all of the informationconveyed over networks, in which case, to the extent information (e.g.,content) is said to be supplied or otherwise provided, the informationmay be provided by sending instructions to retrieve that informationfrom a content delivery network.

Unless specifically stated otherwise, as apparent from the discussion,it is appreciated that throughout this specification discussionsutilizing terms such as “processing,” “computing,” “calculating,”“determining” or the like refer to actions or processes of a specificapparatus, such as a special purpose computer or a similar specialpurpose electronic processing/computing device.

The reader should appreciate that the present application describesseveral inventions. Rather than separating those inventions intomultiple isolated patent applications, applicants have grouped theseinventions into a single document because their related subject matterlends itself to economies in the application process. But the distinctadvantages and aspects of such inventions should not be conflated. Insome cases, embodiments address all of the deficiencies noted herein,but it should be understood that the inventions are independentlyuseful, and some embodiments address only a subset of such problems oroffer other, unmentioned benefits that will be apparent to those ofskill in the art reviewing the present disclosure. Due to costsconstraints, some inventions disclosed herein may not be presentlyclaimed and may be claimed in later filings, such as continuationapplications or by amending the present claims. Similarly, due to spaceconstraints, neither the Abstract nor the Summary of the Inventionsections of the present document should be taken as containing acomprehensive listing of all such inventions or all aspects of suchinventions.

It should be understood that the description and the drawings are notintended to limit the invention to the particular form disclosed, but tothe contrary, the intention is to cover all modifications, equivalents,and alternatives falling within the spirit and scope of the presentinvention as defined by the appended claims.

Modifications and alternative embodiments of various aspects of theinvention will be apparent to those skilled in the art in view of thisdescription. Accordingly, this description and the drawings are to beconstrued as illustrative only and are for the purpose of teaching thoseskilled in the art the general manner of carrying out the invention. Itis to be understood that the forms of the invention shown and describedherein are to be taken as examples of embodiments. Elements andmaterials may be substituted for those illustrated and described herein,parts and processes may be reversed or omitted, certain features may beutilized independently, and embodiments or features of embodiments maybe combined, all as would be apparent to one skilled in the art afterhaving the benefit of this description of the invention. Changes may bemade in the elements described herein without departing from the spiritand scope of the invention as described in the following claims.Headings used herein are for organizational purposes only and are notmeant to be used to limit the scope of the description.

As used throughout this application, the word “may” is used in apermissive sense (i.e., meaning having the potential to), rather thanthe mandatory sense (i.e., meaning must). The words “include”,“including”, and “includes” and the like mean including, but not limitedto. As used throughout this application, the singular forms “a,” “an,”and “the” include plural referents unless the content explicitlyindicates otherwise. Thus, for example, reference to “an” element or “a”element includes a combination of two or more elements, notwithstandinguse of other terms and phrases for one or more elements, such as “one ormore.” The term “or” is, unless indicated otherwise, non-exclusive,i.e., encompassing both “and” and “or.” Terms describing conditionalrelationships, e.g., “in response to X, Y,” “upon X, Y,”, “if X, Y,”“when X, Y,” and the like, encompass causal relationships in which theantecedent is a necessary causal condition, the antecedent is asufficient causal condition, or the antecedent is a contributory causalcondition of the consequent, e.g., “state X occurs upon condition Yobtaining” is generic to “X occurs solely upon Y” and “X occurs upon Yand Z.” Such conditional relationships are not limited to consequencesthat instantly follow the antecedent obtaining, as some consequences maybe delayed, and in conditional statements, antecedents are connected totheir consequents, e.g., the antecedent is relevant to the likelihood ofthe consequent occurring. Statements in which a plurality of attributesor functions are mapped to a plurality of objects (e.g., one or moreprocessors performing steps A, B, C, and D) encompasses both all suchattributes or functions being mapped to all such objects and subsets ofthe attributes or functions being mapped to subsets of the attributes orfunctions (e.g., both all processors each performing steps A-D, and acase in which processor 1 performs step A, processor 2 performs step Band part of step C, and processor 3 performs part of step C and step D),unless otherwise indicated. Further, unless otherwise indicated,statements that one value or action is “based on” another condition orvalue encompass both instances in which the condition or value is thesole factor and instances in which the condition or value is one factoramong a plurality of factors. Unless otherwise indicated, statementsthat “each” instance of some collection have some property should not beread to exclude cases where some otherwise identical or similar membersof a larger collection do not have the property, i.e., each does notnecessarily mean each and every.

To the extent certain U.S. patents, U.S. patent applications, or othermaterials (e.g., articles) have been incorporated by reference, the textof such U.S. patents, U.S. patent applications, and other materials isonly incorporated by reference to the extent that no conflict existsbetween such material and the statements and drawings set forth herein.In the event of such conflict, any such conflicting text in suchincorporated by reference U.S. patents, U.S. patent applications, andother materials is specifically not incorporated by reference herein.

The description of the present application has been presented forpurposes of illustration and description, and is not intended to beexhaustive or limiting of the invention in the form disclosed. Manymodifications and variations will be apparent to those of ordinary skillin the art. Thus, it will be apparent to one skilled in the art thatmodifications may be made as described without departing from the scopeof the claims set out below.

1. A method, comprising: determining, by a hardware computer and basedon an etch bias model, an etch bias for a pattern to be etched using anetch step of a patterning process, the etch bias model comprising aformula including a variable associated with a spatial property of thepattern or with an etch plasma species concentration of the etch step,and including a mathematical term comprising a natural exponentialfunction to the power of a parameter that is fitted or based on an etchtime of the etch step; and adjusting the patterning process based on thedetermined etch bias.
 2. The method of claim 1, wherein the parameter ofthe exponential function is fitted or based on the etch time and areaction constant for the etch step.
 3. The method of claim 1, whereinthe variable comprises the spatial property of the pattern and thespatial property of the pattern is an initial pattern element dimension.4. The method of claim 1, wherein the variable comprises the spatialproperty of the pattern and the formula comprises the variablemultiplied by the mathematical term.
 5. The method of claim 1, whereinthe variable comprises the spatial property of the pattern and theformula comprises a form of CD₀(e^(kt)−1), where CD₀ is the variable andcorresponds to a dimension of the pattern and kt is the parameter thatis fitted for the etch time t of the etch step, wherein k is a reactionconstant for the etch step.
 6. The method of claim 1, wherein thevariable comprises the etch plasma species concentration and the formulafurther comprises a calibration parameter.
 7. The method of claim 1,wherein the variable comprises the etch plasma species concentration andthe formula comprises the variable multiplied by the mathematical term.8. The method of claim 1, wherein the variable comprises the etch plasmaspecies concentration wherein the formula comprises a form ofk₁C_(T0)(e^(kt)−1), wherein k₁ is a calibration parameter, C_(TO) is thevariable and corresponds to the etch plasma species concentration, andk_(t) is the parameter that is fitted for the etch time t of the etchstep, wherein k is a reaction constant for the etch step.
 9. The methodof claim 1, wherein the variable comprises the etch plasma speciesconcentration and the etch plasma species concentration is defined foran etched area of the pattern within an etched material ambitsurrounding an evaluation point on the pattern, wherein the etch plasmaspecies concentration is proportional to the etched area.
 10. The methodof claim 1, wherein the variable comprises the etch plasma speciesconcentration and the formula contains a modified form of the Arrheniusequation incorporated in the power of the exponential function.
 11. Themethod of claim 1, wherein the variable comprises the etch plasmaspecies concentration and the formula incorporates a patterning materialconcentration in the power of the exponential function.
 12. The methodof claim 1, wherein the variable comprises the etch plasma speciesconcentration and wherein the formula has a form comprising$k_{1}{C_{T\; 0}\left( {e^{{Ate}^{\frac{- {CR}}{s}}} - 1} \right)}$where k₁ is a calibration parameter, C_(TO) is the variable andcorresponds to the etch plasma species concentration, C_(R) is apatterning material concentration, At is the parameter that is fittedfor the etch time t of the etch step, wherein A is a frequency factorfor the reaction of the etch step and s is a constant for the etch step.13. The method of claim 1, further comprising: collecting, at each of aplurality of sites in a pattern, a value of a spatial property of thepattern; and fitting, by a hardware computing device and using thevalues of the spatial property, the formula to generate the parameter.14. The method of claim 1, wherein adjusting the patterning processcomprises adjusting a border of a region of the patterning deviceaccording to the calculated etch bias.
 15. A computer program productcomprising a non-transitory computer readable medium having instructionstherein, the instructions, upon execution by a computer system,configured to cause the computer system to at least: determine, based onan etch bias model, an etch bias for a pattern to be etched using anetch step of a patterning process, the etch bias model comprising aformula including a variable associated with a spatial property of thepattern or with an etch plasma species concentration of the etch step,and including a mathematical term comprising a natural exponentialfunction to the power of a parameter that is fitted or based on an etchtime of the etch step; and adjust the patterning process based on thedetermined etch bias.
 16. The computer program product of claim 15,wherein the parameter of the exponential function is fitted or based onthe etch time and a reaction constant for the etch step.
 17. Thecomputer program product of claim 15, wherein the variable comprises thespatial property of the pattern and the spatial property of the patternis an initial pattern element dimension.
 18. The computer programproduct of claim 15, wherein the variable comprises the spatial propertyof the pattern and the formula comprises the variable multiplied by themathematical term.
 19. A method, comprising: determining, by a hardwarecomputer and based on an etch bias model, an etch bias for a pattern tobe etched using an etch step of a patterning process, the etch biasmodel comprising a function of an etch plasma species concentration anda patterning material concentration; and adjusting the patterningprocess based on the determined etch bias.
 20. A computer programproduct comprising a non-transitory computer readable medium havinginstructions therein, the instructions, upon execution by a computersystem, configured to cause the computer system to at least perform themethod of claim 19.